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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: 80242.. x0.
Assume H1: 80242.. x1.
Assume H2: 40dde.. (e4431.. x0) (λ x2 . prim1 x2 x0) (e4431.. x1) (λ x2 . prim1 x2 x1).
Let x2 of type ο be given.
Assume H3: ∀ x3 . 80242.. x3 ⟶ prim1 (e4431.. x3) (d3786.. (e4431.. x0) (e4431.. x1)) ⟶ SNoEq_ (e4431.. x3) x3 x0 ⟶ SNoEq_ (e4431.. x3) x3 x1 ⟶ 099f3.. x0 x3 ⟶ 099f3.. x3 x1 ⟶ nIn (e4431.. x3) x0 ⟶ prim1 (e4431.. x3) x1 ⟶ x2.
Assume H4: prim1 (e4431.. x0) (e4431.. x1) ⟶ SNoEq_ (e4431.. x0) x0 x1 ⟶ prim1 (e4431.. x0) x1 ⟶ x2.
Assume H5: prim1 (e4431.. x1) (e4431.. x0) ⟶ SNoEq_ (e4431.. x1) x0 x1 ⟶ nIn (e4431.. x1) x0 ⟶ x2.
Claim L6: ...
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Claim L7: ...
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Apply unknownprop_1c12738cd89f8c615a541c15b6797bba2a5be97ab5e514c9fd76b3fef06e2aa9 with e4431.. x0, e4431.. x1, λ x3 . prim1 x3 x0, λ x3 . prim1 x3 x1, x2 leaving 4 subgoals.
The subproof is completed by applying H2.
Assume H8: PNoLt_ (d3786.. (e4431.. x0) (e4431.. x1)) (λ x3 . prim1 x3 x0) (λ x3 . prim1 x3 x1).
Apply PNoLt_E_ with d3786.. (e4431.. x0) (e4431.. x1), λ x3 . prim1 x3 x0, λ x3 . prim1 x3 x1, x2 leaving 2 subgoals.
The subproof is completed by applying H8.
Let x3 of type ι be given.
Assume H9: prim1 x3 (d3786.. (e4431.. x0) (e4431.. x1)).
Assume H10: PNoEq_ x3 (λ x4 . prim1 x4 x0) (λ x4 . prim1 x4 x1).
Assume H11: nIn x3 x0.
Assume H12: prim1 x3 x1.
Apply unknownprop_1ac99d32a7ae5dc08fd640ea6c8b661df6b3535fe85e88b30b17c3701cb4c7ce with e4431.. x0, e4431.. x1, x3, x2 leaving 2 subgoals.
The subproof is completed by applying H9.
Assume H13: prim1 x3 (e4431.. x0).
Assume H14: prim1 x3 (e4431.. x1).
Claim L15: ...
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Claim L16: ...
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Claim L17: ...
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Apply unknownprop_2c27682abc32e1c8a2160a07043a81afe1f256df4f0472edf3cf11efdcea8609 with 09072.. x3 (λ x4 . prim1 x4 x0), x2 leaving 2 subgoals.
The subproof is completed by applying L17.
Assume H18: ordinal (e4431.. (09072.. x3 (λ x4 . prim1 x4 x0))).
Assume H19: 1beb9.. (e4431.. (09072.. x3 (λ x4 . prim1 x4 x0))) (09072.. x3 (λ x4 . prim1 x4 x0)).
Claim L20: ...
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Claim L21: ...
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Claim L22: ...
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Apply H3 with 09072.. x3 (λ x4 . prim1 x4 x0) leaving 8 subgoals.
The subproof is completed by applying L17.
Apply L20 with λ x4 x5 . prim1 x5 (d3786.. (e4431.. x0) (e4431.. x1)).
The subproof is completed by applying H9.
Apply L20 with λ x4 x5 . SNoEq_ x5 (09072.. x3 (λ x6 . prim1 x6 x0)) x0.
The subproof is completed by applying L21.
Apply L20 with λ x4 x5 . SNoEq_ x5 (09072.. x3 (λ x6 . prim1 x6 x0)) x1.
The subproof is completed by applying L22.
Apply L20 with λ x4 x5 . 40dde.. (e4431.. x0) (λ x6 . prim1 x6 x0) x5 (λ x6 . prim1 x6 (09072.. x3 (λ x7 . prim1 x7 x0))).
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